Chapter 10 - Stochastic Processes and Stochastic Integration in Banach Spaces

Abstract

This chapter examines the stochastic processes and stochastic integration in Banach spaces. The stochastic integral in Banach spaces is developed with the aid of a vector bilinear integral. The chapter provides convergence theorems and has applied them to establish Ito's formula, the essential tool used in stochastic calculus. The summable processes in the theory play the role of the square integrable martingales in the classical theory. It turns out that every Hilbert valued square integrable martingale is summable. The advantage and purpose of establishing a Lebesgue space for the bilinear vector integral is the possibility to examine weak completeness and weak compactness. The regularity and the Doob-Meyer decomposition of abstract quasimartingales are elaborated in the chapter. One of the main theorems concerning the existence of cadlag modifications of a quasimartingale is presented in the chapter. It is found that for each stopping time, the stopped process is a quasimartingale on satisfying the regularity condition. The existence of right continuous or cadlag modifications is also examined in the chapter.